General Relativity from the three dimensional linear group

Abstract

This letter describes a novel derivation of general relativity by considering the (non)self-consistency of theories whose Hamiltonians are constraints. The constraints, from Hamilton's equations, generate the evolution, while the evolution, in turn, must preserve the constraints. This closure requirement can be used as a selection mechanism for general relativity starting from a very simple set of assumptions. The configuration space is chosen to be a family of $3 \times 3$ positive definite symmetric matrices on some bare 3-manifold. A general Hamiltonian is constructed on this space of matrices which consists of a single constraint per space point. It is assumed that this constraint looks like an energy balance relationship. It will be the sum of a `kinetic' term which is quadratic and undifferentiated in the momenta, and a `potential' term, which is any function of the configuration variables. Further, the constraint must be a scalar under the linear group, the natural symmetry group of the configuration space. This inexorably leads to the ADM Hamiltonian for general relativity. Both the space of Riemannian geometries (Wheeler's superspace), and spacetime are emergent quantities in this analysis.